Hearing shapes of drums - mathematical and physical aspects of isospectrality
arXiv:1101.1239 · doi:10.1103/RevModPhys.82.2213
Abstract
In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.
42 pages, 60 figures
References in corpus (22)
- Random Matrix Theories in Quantum Physics: Common Concepts
- Can One Hear the Shape of a Graph?
- Nodal domains statistics - a criterion for quantum chaos
- Periodic Orbit Theory of Diffraction
- Quantum Phase Extraction in Isospectral Electronic Nanostructures
- Diffractive orbits in quantum billiards
- Inverse spectral problem for analytic plane domains I: Balian-Bloch trace formula
- Resolving isospectral "drums" by counting nodal domains
- Linear Representations and Isospectrality with Boundary Conditions
- Isospectrality in Chaotic Billiards
- Mode fluctuations as fingerprint of chaotic and non-chaotic systems
- Isospectral domains with mixed boundary conditions
- Chaos and Quantum Mechanics
- Tetra and Didi, the cosmic spectral twins
- Numerical investigation of iso-spectral cavities built from triangles
- Exact scattering theory for any straight reflectors in two dimensions
- Penumbra diffraction in the quantization of concave billiards
- Remarks on "Resolving isospectral `drums' by counting nodal domains"
- Diffractive orbits in isospectral billiards
- Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard
- Finite geometries and diffractive orbits in isospectral billiards
- Coincidence of length spectra does not imply isospectrality
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- High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences
- A Convexity Result in the Spectral Geometry of Conformally Equivalent Metrics on Surfaces
- Astroseismology of neutron stars from gravitational waves in the limit of perfect measurement
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